On phase at a resonance in slow-fast Hamiltonian systems

The paper rigorously derives the phase-at-resonance formula for a slow-fast Hamiltonian system with one fast angle, establishing under Assumptions A–E that, outside an exceptional set, the arrival time satisfies τe = τ* + O(√ε ln ν) and the main identity ϕe + H̃1(I0,ϕe,y*,x*)/b(y*,x*) = ϕ0 + (1/ε)∫_0^{τ*,a}(ω0(J0,ηa,ξa)+εω1(J0,ηa,ξa)) dτ + O(√ε ln ν), with O(√ε) when F has no critical points. This is stated and proved via an auxiliary near-resonant first integral E and careful integral estimates (e.g., (2.2), (4.1)–(4.6), (7.3)–(7.11), Lemmas in Secs. 7 and 10) . The candidate’s solution reproduces the same asymptotic relation using a different proof sketch: first-order Lie-transform averaging away from resonance (matching the paper’s (4.1)–(4.5)), a variation-of-constants identity for the (transformed) fast angle, an O(√ε|ln ν|) inner layer around resonance, and a standard ‘scattering-on-resonance’ boundary-layer matching which yields the H̃1/b boundary term. Timing and error scalings agree with the paper’s Proposition 1 (and its refinement when F has no critical points) . While the model omits the paper’s detailed integral lemmas and instead appeals to well-known resonant normal form results for the boundary correction, no substantive conflict arises with the paper; the two arguments are methodologically different but consistent.